Instabilities of a family of oblate stellar spheroids

We have examined the stability of a sequence of oblate elliptical galaxymodels having the Stackel form suggested by Kuz'min & Kutuzov. We have employedthe 2-integral DFs given by Dejonghe & de Zeeuw for which flattenednon-rotating models are characterized by counter-streaming motion and areradially cool; we introduce net rotation in some models by changing the sign ofthe z-component of angular momentum for a fraction of the particles. We havefound that all non-rotating and slowly rotating members of this sequencerounder than E7 are stable, and that even maximally rotating models rounderthan E4 are stable. In the absence of strong rotation, the most disruptiveinstability, and the last to be stabilized by increasing thickness, is alopsided (m=1) mode. This instability appears to be driven by counter-rotationin radially cool models. Its vigour is lessened as rotation is increased, butit remains strong even in models with net angular momentum 90% of that of amaximally rotating model before finally disappearing in maximally rotatingmodels. Strongly rotating models are more unstable to bar-forming modes whichafflict maximally rotating models with c/a >~ 0.5, but this mode is quicklystabilized by moderate fractions of counter-rotating particles. Bendinginstabilities appear not to be very important; they are detectable in the innerparts of the flatter models, but are less vigorous and more easily stabilizedthan the lop-sided or bar modes in every case. We briefly discuss the possiblerelevance of the lop-sided instability to the existence of many lop-sided diskgalaxies.Comment: 18 pages including 8 figures, TeX, macro file included, to appear in Monthly Notices of RA